Historical remarks and conclusion

A few concluding remarks regarding the history of
the wall formula are in order.
It has been known since its inception that the naive wall formula gives
unphysical answers in the case of constant-velocity
translations and rotations. This was first regarded
as a kinetic gas `drift' effect [29].
It should be noted that the recipe presented in [29],
namely to subtract this drift component,
is equivalent in practice to the recipe (4.9) that we have
presented here, provided we ignore dilations.
It is also important to realize that the
argumentation in [29] for this subtraction
appears to be ad hoc, being based on a
`least-structured drift pattern' reasoning.
A stated condition on this subtraction
was that the resulting deformation preserve the location
of the `center of mass' (centroid) of the cavity,
for reasons particular to the nuclear application [29].
This condition seems to have become standard practice in
numerical tests of the wall formula [158,149,30,31,32].
However, as Fig. 4.7a shows,
this condition is generally not equivalent to the above subtraction
of translation and rotation components 4.1.
This seems to invalidate the theorem presented in Section 7.1 of [29].
Where the flaw in their reasoning lies we are not sure.

The consideration of the special nature of
dilations is absent from the literature.
Even if we restrict ourselves to volume-preserving deformations
(the nuclear dissipation case),
then deformations of certain cavities can be found for which
the dilation correction is significant.
This correction can only be large if the cavity has a large
variation in radius (i.e. is highly non-spherical).
We illustrate this in Fig. 4.7b.
We suggest this as a possible reason why major discrepancies
due to dilation
have not emerged
in the
numerical tests of the wall formula until now.
Such tests have generally been of shapes close to a 3D sphere
[29,158,149,30,31,32].

Hence we believe that the new recipe presented,
along with the associated theory and in conjunction with
the particular power-law dependences from the previous
chapter, is a significant
step in the treatment of one-body dissipation
and of dissipation in -dimensional cavities in general.